Myswizard

(This is another of the many new Food For Thought articles in the upcoming new book, “There Are No Secrets.”)

As we travel the path to enlightenment, relationships remain an important part of our spiritual journey. Although enlightenment has been associated with leaving the world of materialism and even form, devotion to God is becoming an intrinsic part of many relationships. Even within relationships where only one may be on the path, it is important to stay within the field of spiritual alignment. Any deviations may have a tendency to keep one from forging ahead. To present what alignment and non-alignment look and feel like, creates an opportunity for higher evolution within the relationship.

That which is consciously aligned is congruent with the same (like) or higher ideologies. Alignment does not force change, but is wise enough to see and allow, evolving as it learns. Alignment does not control nor is it afraid of loss. It is not overstimulated or overemotional. It is calm and aware without hidden agendas. It has the well-being of the “other” in mind at all times. It is not jealous, scheming, cynical, or judgmental. An aligned individual conveys their feelings within the context of higher consciousness, and is willing to make things work. When ego is transcended, alignment is a commonality of higher levels. Alignment and higher calibrations of consciousness go hand in hand.*

Non-alignment is argumentative, ego driven, victimizing, blaming, temperamental, and controlling. Non-alignment doesn’t feel right. What it creates are feelings of fear and uneasiness. Within all types of relationships, non-alignment feels strained, uneasy, unsettling and forced. It is a negative (or oppressive) attractor field, being a consequence of lower levels of consciousness. The spiritually non-aligned at the core level of the ego, do not have the capacity for loving and consciously congruent long term relationships.

In relationships where one is spiritually aligned, there can be commonalities, however it takes both individuals to create total alignment. Non-aligned relationships have been known to stay together for other reasons. This doesn’t necessarily mean things aren’t working. It simply is not optimally aligned. To seek our perception of perfection within all relationships may be an exercise in futility. When there is complete trust, kindness, and love however, alignment is present.

These are a few hints as to whether or not a relationship (any relationship) is working. To be spiritually aligned with another may not always be perfection, but it’s a beautiful work in progress. That is what spiritual alignment is; progress with the mutual intention to live within a higher mode of consciousness.
*See Map of Consciousness(referential) and Devotional Nonduality links:(Consciousnessproject.org)

©Myswizard all rights reserved ‘05-’06

In dynamical systems, an attractor is a set to which the system evolves after a long enough time. For the set to be an attractor, trajectories that get close enough to the attractor must remain close even if slightly disturbed. Geometrically, an attractor can be a point, a curve, a manifold, or even a complicated set with fractal structures known as a strange attractor. Describing the attractors of chaotic dynamical systems has been one of the achievements of chaos theory.

A trajectory of the dynamical system in the attractor does not have to satisfy any special constraints except for remaining on the attractor. The trajectory may be periodic or chaotic or of any other type.

Motivation and definition
Dynamical systems are often described in terms of differential equations. These equations describe the behavior of the system for a short period of time. To determine the behavior for longer periods it is necessary to integrate the equations, either through analytical means or through iteration, often with the aid of computers. Dynamical systems that come from applications tend to be dissipative: if it were not for some driving force the motion would cease. (The dissipation may come from internal friction, thermodynamic losses, or loss of material, among many causes.) The dissipation and the driving force tend to combine to kill out initial transients and settle the system into its typical behavior. The part of the phase space of the dynamical system corresponding to the typical behavior is the attracting set or attractor.

Invariant sets and limit sets are similar to the attractor concept. An invariant set is a set that evolves to itself under the dynamics. Attractors may contain invariant sets. A limit set are the states a system goes to after an infinite amount of time. Attractors are limit sets, but not all limit sets are attractors. It is possible to have a system converge to a limit set, but if placed in the limit set, have small perturbations that knock it off to never return.

As an example, the damped pendulum has two invariant points: the point x0 of minimum height and the point x1 of maximum height. The point x0 is also a limit set, as trajectories converge to it; the point x1 is not a limit set. Because of the dissipation, the point x0 is also an attractor. If there were no dissipation, x0 would not be an attractor.

Mathematical definition
In a dynamical system with dynamics f(t, •), the attractor Λ is a subset of the phase space such that:

there is a neighborhood of Λ, the attraction basin, that will converge to any open set containing Λ, and
f(t, Λ) ⊃ Λ for large t.
The open subset condition assures that phase space points in the neighborhood of the attractor converge to it.

Types of attractors
Attractors are parts of the phase space of the dynamical system. Until the 1960s, as evidenced by textbooks of that era, attractors were thought of as being geometrical subsets of the phase space: points, lines, surfaces, volumes. The (topologically) wild sets that had been observed were thought to be fragile anomalies. Stephen Smale was able to show that his horseshoe map was robust and that its attractor had the structure of a Cantor set.

Two simple attractors are the fixed point and the limit cycle. There can be many other geometrical sets that are attractors. When these sets (or the motions on them), are hard to describe, then the attractor is a strange attractor, as described in the section below.

Fixed point
A fixed point is a point that a system evolves towards, such as the final states of a falling pebble, a damped pendulum, or the water in a glass. It corresponds to a fixed point of the evolution function that is also attracting.

Limit cycle
A limit cycle is a periodic orbit of the system that is isolated. Examples include the swings of a pendulum clock, the tuning circuit of a radio, and the heartbeat while resting. The ideal pendulum is not an example because its orbits are not isolated. In phase space of the ideal pendulum, near any point of a periodic orbit there is another point that belongs to a different periodic orbit.

Limit tori
There may be more than one frequency in the periodic trajectory of the system through the state of a limit cycle. If two of these frequencies form an irrational fraction(i.e. they are incommensurate), the trajectory will no longer be closed, and the limit cycle becomes a limit torus. We call this kind of attractor Nt-torus if there are Nt incommensurate frequencies. For example it is a 2-torus:

A time series corresponding to this attractor is a quasiperiodic series: A discretely sampled sum of Nt periodic functions (not necessarily sine waves) with incommensurate frequencies. Such a time series does no longer have a strict periodicity, but its power spectrum still consists only of sharp lines.

Strange attractor
A plot of Lorenz’s strange attractor for values ρ=28, σ = 10, β = 8/3An attractor is informally described as strange if it has non-integer dimension or if the dynamics on the attractor are chaotic. The term was coined by David Ruelle and Floris Takens to describe the attractor that resulted from a series of bifurcations of a system describing fluid flow. Strange attractors are often differentiable in a few directions and like a Cantor dust (and therefore not differentiable) in others.

The Hénon attractor, Rössler attractor, and the Lorenz attractor are examples of strange attractors.

Partial differential equations
Parabolic partial differential equations may have finite-dimensional attractors. The diffusive part of the equation damps higher frequencies and in some cases leads to a global attractor. The Ginzburg-Landau, the Kuramoto-Sivashinsky, and the two-dimensional, forced Navier-Stokes equations are all known to have global attractors of finite dimension.

For the three-dimensional, incompressible Navier-Stokes equation with periodic boundary conditions, if it has a global attractor, then this attractor will be of finite dimension.

Further reading
Edward N. Lorenz (1996) The Essence of Chaos ISBN 0295975148
James Gleick (1988) Chaos: Making a New Science ISBN 02959751

This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article “Attractor”.

In mathematics and physics, chaos theory describes the behavior of certain nonlinear dynamical systems that under certain conditions exhibit a phenomenon known as chaos. Among the characteristics of chaotic systems, described below, is sensitivity to initial conditions (popularly referred to as the butterfly effect). As a result of this sensitivity, the behavior of systems that exhibit chaos appears to be random, even though the system is deterministic in the sense that it is well defined and contains no random parameters. Examples of such systems include the atmosphere, the solar system, plate tectonics, turbulent fluids, economics, and population growth.

Systems that exhibit mathematical chaos are deterministic and thus orderly in some sense; this technical use of the word chaos is at odds with common parlance, which suggests complete disorder. (See the article on mythological chaos for a discussion of the origin of the word in mythology, and other uses.) A related field of physics called quantum chaos theory studies non-deterministic systems that follow the laws of quantum mechanics.

Chaotic dynamics
For a dynamical system to be classified as chaotic, most scientists will agree that it must have the following properties:
.it must be sensitive to initial conditions,
.it must be topologically mixing, and
.its periodic orbits must be dense.
Sensitivity to initial conditions means that each point in such a system is arbitrarily closely approximated by other points with significantly different future trajectories. Thus, an arbitrarily small perturbation of the current trajectory may lead to significantly different future behavior.

Sensitivity to initial conditions is popularly known as the “butterfly effect”, suggesting that the flapping of a butterfly’s wings might create tiny changes in the atmosphere, which could over time cause a tornado to occur. The flapping wing represents a small change in the initial condition of the system, which causes a chain of events leading to large-scale phenomena. Had the butterfly not flapped its wings, the trajectory of the system might have been vastly different.

Sensitivity to initial conditions is often confused with chaos in popular accounts. It can also be a subtle property, since it depends on a choice of metric, or the notion of distance in the phase space of the system. For example, consider the simple dynamical system produced by repeatedly doubling an initial value (defined by the mapping on the real line from x to 2x). This system has sensitive dependence on initial conditions everywhere, since any pair of nearby points will eventually become widely separated. However, it has extremely simple behavior, as all points except 0 tend to infinity. If instead we use the bounded metric on the line obtained by adding the point at infinity and viewing the result as a circle, the system no longer is sensitive to initial conditions. For this reason, in defining chaos, attention is normally restricted to systems with bounded metrics, or closed, bounded invariant subsets of unbounded systems.

Even for bounded systems, sensitivity to initial conditions is not identical with chaos. For example, consider the two-dimension torus described by a pair of angles (x,y), each ranging between zero and 2π. Define a mapping that takes any point (x,y) to (2x, y + a), where a is any number such that a/2π is irrational. Because of the doubling in the first coordinate, the mapping exhibits sensitive dependence on initial conditions. However, because of the irrational rotation in the second coordinate, there are no periodic orbits, and hence the mapping is not chaotic according to the definition above.

Topologically mixing means that the system will evolve over time so that any given region or open set of its phase space will eventually overlap with any other given region. Here, “mixing” is really meant to correspond to the standard intuition: the mixing of colored dyes or fluids is an example of a chaotic system.

Attractors
Some dynamical systems are chaotic everywhere (see e.g. Anosov diffeomorphisms) but in many cases chaotic behavior is found only in a subset of phase space. The cases of most interest arise when the chaotic behavior takes place on an attractor, since then a large set of initial conditions will lead to orbits that converge to this chaotic region.

An easy way to visualize a chaotic attractor is to start with a point in the basin of attraction of the attractor, and then simply plot its subsequent orbit. Because of the topological transitivity condition, this is likely to produce a picture of the entire final attractor.

phase diagram for a damped driven pendulum, with double period motion

For instance, in a system describing a pendulum, the phase space might be two-dimensional, consisting of information about position and velocity. One might plot the position of a pendulum against its velocity. A pendulum at rest will be plotted as a point, and one in periodic motion will be plotted as a simple closed curve. When such a plot forms a closed curve, the curve is called an orbit. Our pendulum has an infinite number of such orbits, forming a pencil of nested ellipses about the origin.

Strange attractors
While most of the motion types mentioned above give rise to very simple attractors, such as points and circle-like curves called limit cycles, chaotic motion gives rise to what are known as strange attractors, attractors that can have great detail and complexity. For instance, a simple three-dimensional model of the Lorenz weather system gives rise to the famous Lorenz attractor. The Lorenz attractor is perhaps one of the best-known chaotic system diagrams, probably because not only was it one of the first, but it is one of the most complex and as such gives rise to a very interesting pattern which looks like the wings of a butterfly. Another such attractor is the Rössler Map, which experiences period-two doubling route to chaos, like the logistic map.

Strange attractors occur in both continuous dynamical systems (such as the Lorenz system) and in some discrete systems (such as the Hénon map). Other discrete dynamical systems have a repelling structure called a Julia set which forms at the boundary between basins of attraction of fixed points - Julia sets can be thought of as strange repellers. Both strange attractors and Julia sets typically have a fractal structure.

The Poincaré-Bendixson theorem shows that a strange attractor can only arise in a continuous dynamical system if it has three or more dimensions. However, no such restriction applies to discrete systems, which can exhibit strange attractors in two or even one dimensional systems.

The initial conditions of three or more bodies interacting through gravitational attraction (see the n-body problem) can be arranged to produce chaotic motion.

History
The first discoverer of chaos can plausibly be argued to be Jacques Hadamard, who in 1898 published an influential study of the chaotic motion of a free particle gliding frictionlessly on a surface of constant negative curvature. In the system studied, Hadamard’s billiards, Hadamard was able to show that all trajectories are unstable, in that all particle trajectories diverge exponentially from one-another, with positive Lyapunov exponent. In the early 1900s, Henri Poincaré while studying the three-body problem, found that there can be orbits which are nonperiodic, and yet not forever increasing nor approaching a fixed point. Much of the early theory was developed almost entirely by mathematicians, under the name of ergodic theory. Later studies, also on the topic of nonlinear differential equations, were carried out by G.D. Birkhoff, A.N. Kolmogorov, M.L. Cartwright, J.E. Littlewood, and Stephen Smale. Except for Smale, these studies were all directly inspired by physics: the three-body problem in the case of Birkhoff, turbulence and astronomical problems in the case of Kolmogorov, and radio engineering in the case of Cartwright and Littlewood. Although chaotic planetary motion had not been observed, experimentalists had encountered turbulence in fluid motion and nonperiodic oscillation in radio circuits without the benefit of a theory to explain what they were seeing.

Chaos theory progressed more rapidly after mid-century, when it first became evident for some scientists that linear theory, the prevailing system theory at that time, simply could not explain the observed behavior of certain experiments like that of the logistic map. The main catalyst for the development of chaos theory was the electronic computer. Much of the mathematics of chaos theory involves the repeated iteration of simple mathematical formulas, which would be impractical to do by hand. Electronic computers made these repeated calculations practical. One of the earliest electronic digital computers, ENIAC, was used to run simple weather forecasting models.

An early pioneer of the theory was Edward Lorenz whose interest in chaos came about accidentally through his work on weather prediction in 1961. Lorenz was using a basic computer, a Royal McBee LGP-30, to run his weather simulation. He wanted to see a sequence of data again and to save time he started the simulation in the middle of its course. He was able to do this by entering a printout of the data corresponding to conditions in the middle of his simulation which he had calculated last time.

To his surprise the weather that the machine began to predict was completely different from the weather calculated before. Lorenz tracked this down to the computer printout. The printout rounded variables off to a 3-digit number, but the computer worked with 6-digit numbers. This difference is tiny and the consensus at the time would have been that it should have had practically no effect. However Lorenz had discovered that small changes in initial conditions produced large changes in the long-term outcome.

Yoshisuke Ueda independently identified a chaotic phenomenon as such by using an analog computer on November 27, 1961. The chaos exhibited by an analog computer is truly a natural phenomenon, in contrast with those discovered by a digital computer. Ueda’s supervising professor, Hayashi, did not believe in chaos throughout his life, and thus he prohibited Ueda from publishing his findings until 1970.

The term chaos as used in mathematics was coined by the applied mathematician James A. Yorke.

The availability of cheaper, more powerful computers broadens the applicability of chaos theory. Currently, chaos theory continues to be a very active area of research.

Mathematical theory
Sarkovskii’s theorem is the basis of the Li and Yorke (1975) proof that any one-dimensional system which exhibits a regular cycle of period three will also display regular cycles of every other length as well as completely chaotic orbits.

Mathematicians have devised many additional ways to make quantitative statements about chaotic systems. These include: fractal dimension of the attractor, Lyapunov exponents, recurrence plots, Poincaré maps, bifurcation diagrams, Transfer operator

Minimum complexity of a chaotic system
Bifurcation diagram of a logistic map, displaying chaotic behavior past a threshold

Simple systems can also produce chaos without relying on differential equations. An example is the logistic map, which is a difference equation (recurrence relation) that describes population growth over time.

Even the evolution of simple discrete systems, such as cellular automata, can heavily depend on initial conditions. Stephen Wolfram has investigated a cellular automaton with this property, termed by him rule 30.

A minimal model for conservative (reversible) chaotic behavior is provided by Arnold’s cat map.

Other examples of chaotic systems
Double pendulum
Logistic map
Arnold’s cat map
Hénon map
Lorenz model
Smale horseshoe
Dynamical billiards
Chua’s circuit
Rössler Map
Economic bubble

Application
Chaos theory is applied in many scientific disciplines: mathematics, biology, computer science, economics, engineering, philosophy, physics, politics, population dynamics, psychology, robotics, etc.

This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article “Chaos Theory”.